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Question Number 153870 Answers: 1 Comments: 3

find the minimum and maximum value of (5/(f(θ)+3)) where f(θ)=8cos θ−15 sin θ

$$\mathrm{find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{and}\:\mathrm{maximum}\:\mathrm{value} \\ $$$$\mathrm{of}\:\frac{\mathrm{5}}{{f}\left(\theta\right)+\mathrm{3}}\:\mathrm{where}\:{f}\left(\theta\right)=\mathrm{8cos}\:\theta−\mathrm{15}\:\mathrm{sin}\:\theta \\ $$

Question Number 153864 Answers: 0 Comments: 1

Question Number 153866 Answers: 1 Comments: 0

3+(√(3+(√(6+(√(9+(√(12+…+(√(99))))))))))=?

$$\mathrm{3}+\sqrt{\mathrm{3}+\sqrt{\mathrm{6}+\sqrt{\mathrm{9}+\sqrt{\mathrm{12}+\ldots+\sqrt{\mathrm{99}}}}}}=? \\ $$

Question Number 153862 Answers: 0 Comments: 0

Question Number 153860 Answers: 1 Comments: 0

Question Number 153858 Answers: 2 Comments: 1

Question Number 153857 Answers: 1 Comments: 1

Question Number 153847 Answers: 2 Comments: 0

S = x + 2x^2 + ... + nx^n

$$\boldsymbol{\mathrm{S}}\:=\:\mathrm{x}\:+\:\mathrm{2x}^{\mathrm{2}} \:+\:...\:+\:\mathrm{nx}^{\boldsymbol{\mathrm{n}}} \\ $$$$ \\ $$

Question Number 153840 Answers: 1 Comments: 0

log _e (x)+log _x (e)+log _(((e/x))) (x)=(5/2) x=?

$$\:\:\:\:\mathrm{log}\:_{{e}} \left({x}\right)+\mathrm{log}\:_{{x}} \left({e}\right)+\mathrm{log}\:_{\left(\frac{{e}}{{x}}\right)} \left({x}\right)=\frac{\mathrm{5}}{\mathrm{2}} \\ $$$$\:{x}=? \\ $$

Question Number 153839 Answers: 1 Comments: 1

Question Number 153829 Answers: 0 Comments: 0

L(x) = (6/π^2 )(Li_2 (x) + (1/2) log(x)log(1-x) Find: 𝛀 =∫_( 0) ^( 1) L(x)∙Li_2 (x) dx

$$\mathrm{L}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{6}}{\pi^{\mathrm{2}} }\left(\mathrm{Li}_{\mathrm{2}} \left(\mathrm{x}\right)\:+\:\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{log}\left(\mathrm{x}\right)\mathrm{log}\left(\mathrm{1}-\mathrm{x}\right)\right. \\ $$$$\mathrm{Find}:\:\:\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{L}\left(\mathrm{x}\right)\centerdot\mathrm{Li}_{\mathrm{2}} \left(\mathrm{x}\right)\:\mathrm{dx} \\ $$

Question Number 153842 Answers: 0 Comments: 0

Question Number 153817 Answers: 0 Comments: 3

Question Number 153808 Answers: 0 Comments: 0

If 0<a≤b<1 then: ∫_( a) ^( b) ∫_( a) ^( b) ∫_a ^( b) (((1 - xyz)/(1 + xyz)))^3 dxdydz ≥ (∫_a ^( b) ((1 - x^3 )/(1 + x^3 )) dx)^3

$$\mathrm{If}\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b}<\mathrm{1}\:\:\mathrm{then}: \\ $$$$\underset{\:\boldsymbol{\mathrm{a}}} {\overset{\:\boldsymbol{\mathrm{b}}} {\int}}\underset{\:\boldsymbol{\mathrm{a}}} {\overset{\:\boldsymbol{\mathrm{b}}} {\int}}\underset{\boldsymbol{\mathrm{a}}} {\overset{\:\boldsymbol{\mathrm{b}}} {\int}}\left(\frac{\mathrm{1}\:-\:\mathrm{xyz}}{\mathrm{1}\:+\:\mathrm{xyz}}\right)^{\mathrm{3}} \mathrm{dxdydz}\:\geqslant\:\left(\underset{\boldsymbol{\mathrm{a}}} {\overset{\:\boldsymbol{\mathrm{b}}} {\int}}\frac{\mathrm{1}\:-\:\mathrm{x}^{\mathrm{3}} }{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{3}} }\:\mathrm{dx}\right)^{\mathrm{3}} \\ $$

Question Number 153803 Answers: 1 Comments: 1

solve for x cos^2 x − cos^2 2x = cos^2 4x − cos^2 3x

$${solve}\:{for}\:{x} \\ $$$${cos}^{\mathrm{2}} {x}\:−\:{cos}^{\mathrm{2}} \mathrm{2}{x}\:=\:{cos}^{\mathrm{2}} \mathrm{4}{x}\:−\:{cos}^{\mathrm{2}} \mathrm{3}{x} \\ $$

Question Number 153800 Answers: 2 Comments: 0

Question Number 153784 Answers: 0 Comments: 1

Question Number 153781 Answers: 1 Comments: 1

Question Number 153780 Answers: 1 Comments: 0

⌊ ((125)/(12)) ⌋ =10 or 11 ?

$$\:\lfloor\:\frac{\mathrm{125}}{\mathrm{12}}\:\rfloor\:=\mathrm{10}\:{or}\:\mathrm{11}\:? \\ $$

Question Number 153775 Answers: 1 Comments: 1

Question Number 153772 Answers: 1 Comments: 2

Question Number 153769 Answers: 1 Comments: 0

Question Number 153765 Answers: 2 Comments: 0

Given f:R→R is increasing positive function with lim_(x→∞) ((f(3x))/(f(x)))=1 . What the value of lim_(x→∞) ((f(2x))/(f(x))). (A) 3 (B) (3/2) (C) 1 (D)(2/3) (E) ∞

$$\:{Given}\:{f}:{R}\rightarrow{R}\:{is}\:{increasing}\:{positive} \\ $$$${function}\:{with}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{f}\left(\mathrm{3}{x}\right)}{{f}\left({x}\right)}=\mathrm{1}\:.\: \\ $$$${What}\:{the}\:{value}\:{of}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{f}\left(\mathrm{2}{x}\right)}{{f}\left({x}\right)}. \\ $$$$\left({A}\right)\:\mathrm{3}\:\:\:\:\:\left({B}\right)\:\frac{\mathrm{3}}{\mathrm{2}}\:\:\:\:\:\left({C}\right)\:\mathrm{1}\:\:\:\:\:\left({D}\right)\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\left({E}\right)\:\infty \\ $$

Question Number 153764 Answers: 1 Comments: 2

Prove without any software: ∫_( 0) ^( 1) ∫_( 0) ^( 1) (√(1 - (((x + y)/2))^2 )) dxdy > (π/4)

$$\mathrm{Prove}\:\mathrm{without}\:\mathrm{any}\:\mathrm{software}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\sqrt{\mathrm{1}\:-\:\left(\frac{\mathrm{x}\:+\:\mathrm{y}}{\mathrm{2}}\right)^{\mathrm{2}} }\:\mathrm{dxdy}\:>\:\frac{\pi}{\mathrm{4}} \\ $$

Question Number 153763 Answers: 1 Comments: 0

Determine all pairs (x;y) of integers which satisfy ∣x^2 - y^2 ∣ - (√(16y + 1)) = 0

$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{pairs}\:\left(\mathrm{x};\mathrm{y}\right)\:\mathrm{of}\:\mathrm{integers} \\ $$$$\mathrm{which}\:\mathrm{satisfy} \\ $$$$\mid\mathrm{x}^{\mathrm{2}} \:-\:\mathrm{y}^{\mathrm{2}} \mid\:-\:\sqrt{\mathrm{16y}\:+\:\mathrm{1}}\:=\:\mathrm{0} \\ $$

Question Number 153760 Answers: 1 Comments: 0

∫ sin^2 4x cos 4x dx=

$$\int\:\mathrm{sin}^{\mathrm{2}} \mathrm{4}{x}\:\mathrm{cos}\:\mathrm{4}{x}\:{dx}= \\ $$

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